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We consider a time-dependent quantum linear oscillator coupled to a bath atan arbitrary strength. We then introduce a generalized Jarzynski equality (GJE)which includes the terms reflecting the system-bath coupling. This enables usto study systematically the coupling effect on the linear oscillator in anon-equilibrium process. This is also associated with the second law ofthermodynamics beyond the weak-coupling limit. We next take into considerationthe GJE in the classical limit. By this generalization we show that theJarzynski equality in its original form can be associated with the second law,in both quantal and classical domains, only in the vanishingly small couplingregime.

Jarzynski equality and the second law of thermodynamics beyond the weak-coupling limit: The quantum Brownian oscillator